Place each veal round between 2 pieces of plastic wrap and pound to a 1/4-inch thickness, using a meat mallet or the bottom of a small, heavy pan. Transfer the veal to a baking sheet. Combine the oregano, lemon zest, 2 teaspoons salt and 1 teaspoon pepper in a small bowl, then rub over the veal. Cover in plastic and place in the refrigerator for at least 30 minutes.
Remove the veal from the refrigerator and dredge in flour to coat both sides. Shake to remove excess flour, then set the veal aside on a plate or tray.
In a large skillet, heat 4 tablespoons of the butter over medium heat until melted and bubbling. Working in batches, fry the veal until golden and just cooked through, about 2 minutes per side. Remove from the pan and set aside.
Add the remaining 4 tablespoons butter and the onions to the pan and cook until the onions are softened, 2 to 3 minutes. Add the mushrooms, Marsala, capers, mustard, 1 teaspoon salt and 1 teaspoon pepper, and saute until soft and golden, 3 to 5 minutes. Add the chicken broth and cook until the sauce is reduced by half and thickened.
Transfer the cooked veal to a serving plate and pour the sauce all over it. Top with the parsley and serve.